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DESIGN OF ACHROMATIC AND APOCHROMATIC MICROOBJECTIVES
WITH PLASTIC LENSES
Grigoriy I. Greisukh, Evgeniy G. Ezhov, Il’ya X Levin and Sergei A.
Stepanov
Penza State University of Architecture and Construction, 28
Titov Street, 440028 Penza, Russia.
E-mail: [email protected]
This work shows the possibility and the efficiency of a single diffractive lens usage
to achromatize and apochromatize microobjectives with plastic lenses. It also gives
recommendations on assembling the source schemes of objectives and calculating
the construction parameters required for a following optimization. It was also
shown that achievable optical characteristics of achromatic and apochromatic
microobjectives with plastic lenses satisfy the qualifying standards for cell-phone
objectives and CCTV-cameras.
Introduction
As is well known, the first-order chromatism correction of refractive lens
optical systems within the limited choice of optical materials can be achieved by
means of diffractive lenses (DL) (see [1-5], for example). One of the possible ways
to achromatize or apochromatize an optical system with the chromatism above the
required level is shown in [6]. It implies the usage of diffractive-refractive
correction unit, consisting of DL and one or two refractive lenses (RL). The
correction unit can be build using both additional lenses and own RLs of the
optical system, located near the aperture stop.
The present work gives recommendations for rational assembling the
microobjectives scheme that allow to achieve achromatization or even
apochromatization by means of merely a single additional DL. These
recommendations also open the way for further optimization basing on the
calculated construction parameters. Both the recommendations and the calculations
are given for photo- and cell-phone microobjectives and CCTV-systems of security
cameras working in Day/Night vision, that is in the spectrum interval that includes
visible and near infrared ranges from  min  0,4 μm до  max  0,9 μm.
The low cost requirement and decent optical characteristics of the given
microobjective class stipulate the usage of optical plastic materials to produce the
elements of their optical schemes. The modern shaping methods based on the
precision punching allow replicating acrylic lenses with aspheric refractors and
also allow applying a diffractive microrelief on demand [7,8].
Scheme assembling and achromat optimization results
The analysis of different vendors’ microobjective series of a given class [9] gave
the following default values of their basic optical characteristics: the back focal
length f   3,7 mm; relative aperture – 1:2,4; field angle in object space –
2  =60○. Such characteristics justify the starting scheme of an achromatic lens
based on a triplet having its aspheric lenses made of the same crown-like plastic.
In the visible range, i.e. between blue F- and red C- spectrum lines of hydrogen
(  min   F  0,48613 μm и  max   C  0,65626 μm), it is possible to confine with
achromatization that provides the equality of system image distances at the edges
of the selected spectrum range ( s min  s max ), due to a small width of a secondary
spectrum. In a wider spectrum range that includes visible and near infrared ranges,
the full exclusion of a primary chromatism influence to an objective’s resolution is
only possible via apochromatization. It should provide the equality of system
image distances on three wavelengths min    max , that is: s min  s  s max .
Refractive indexes and Abbe numbers as well as all other optical parameters
in this work are presented for the yellow d-line of helium (  d  0,58756 μm) that
was used for apochromatization alongside with  min =0,4 μm и  max =0,9 μm due
to rigid requirements to the camera’s resolution in the Day vision; that is    d .
As a crown-like optical material we have chosen polymethylmethacrylate (acrylic
PMMA) ( nd  1,491756;  d  57,4408), and as a flint-like – polycarbonate (PC)
( nd  1,585470;  d  29,9092).
To achromatize the triplet with all RLs build of PMMA it is all-sufficient to
include a single DL into the scheme [10]. Figure 1 shows the optical scheme of
such 4-lens achromat and Table 1 lists its construction parameters gained via the
optimization.
Figure 1. Optical scheme of a 4-lens achromat: 1– aperture stop; 2 – DL.
Table 1. Lens Listing for Four - Lens Achromat
Surface
Radius
(mm)
Thickness
(mm)
Aspheric coefficients
Glass
 2 10 ,
 3 10 2 ,
 4 10 2 ,
 5 10 2 ,
mm-3
mm-5
mm-7
mm-9
-2.2695
-6.1431
-3.5615
-16.0129
0.2676
-3.5652
-2.7021
8.5846
2
OBJ:
Infinity
Infinity
Air
1
1.7319
0.7600
PMMA
b
2
3.9729
0.1000
Air
STO:
Infinity
0.3425
Air
4
-1.6791
0.6365
PMMA
2.1514
-21.8536
17.5853
10.6459
5
-1.3310
0.0500
Air
5.3303
3.6462
4.1340
6.7561
6
2.0066
0.7000
PММА
-14.6056
8.4726
-2.5418
0.4269
7
1.5455
2.0556
Air
-18.7189
5.7059
-1.7250
0.0791
IMG:
Infinity
0.0000
a
3.712-mm FL
Diffraction Order and Diffractive lens phase coefficients: m =1; A1  -107 mm-2 ;
b
A2  43.9432 mm-4 ; A3  -493.7187 mm-6 ; A4  2909.2307 mm-8 ;
A5  -7311.6339 mm-10 ; A6  8711.5723 mm-12 ; A7  -4461.2733 mm-14 ;
A8  551.7575 mm-16 .
In Table 1 and further down  i , m и A j  are parameters of aspheric surfaces
and DL’s ring microstructure applied to the back side of the aspheric surface of the
first RL. The above parameters appear in equations that are used to describe optical
elements, in particular, in optical design program ZEMAX [11].
Even aspheric surfaces are described with the following equation:
z
c 2
1  1  c2  2
  i  2 i ,
(1)
i 2
Where z is the coordinate of a surface point that is away from the optic axis to the
length of  in the coordinates, bound to the vertex of this surface, ñ  1 r is a
surface curvature in that vertex,  i where i  1, 2, 3, ... are aspheric coefficients.
The equation of the DL structure spatial frequency is:
(  ) 
1 d
,
2 m d
(2)
where surface adds phase to the ray
  m  Aj  2 j ,
j 1
(3)
 is the distance measured from the optic axis, m is the number of the diffraction
order.
The DL with the structure described with (2) and (3) has the optical power
determined by A1 and m :
   A1 m  .
(4)
A j , where j  2, 3,... , determine the contribution of the DL to the spherical
aberration of 3th, 5th and subsequent orders.
The analysis of aberration curves of a microobjective with the parameters
shown in Table 1 has shown that it is possible to combine achromatization with the
significant reduction of the secondary spectrum by limiting the back focal length in
all visible spectrum range with the value s F  8 μm. It is also possible to
significantly lower monochromatic aberrations and the spherochromatism. As a
result, the polychromatic resolution of an objective is 98 mm-1 across the field
angle 2  60 o with the contrast K =0,5; and with K =0,78 it never lowers below
50 mm-1. The distortion within the entire field of vision does not exceed  0,5%.
However this objective has a notable drawback. It is the quite wide range of
field angles in the image domain that leads to a heterogeneous lighting of the
multielement photodetector surface. Indeed, the main rays fall onto peripheral
  1,12 max , which is 33,6º.
elements of the photodetector under the angle  max
The above drawback can be mostly eliminated by including an additional RL
as a terminal lens. Figure 2 shows the optical scheme of the five-lens achromat,
and Table 2 lists its construction parameters gained via the optimization.
Figure 2. Principal optical scheme of the five-lens achromat: 1– aperture stop; 2 –
DL.
Table 2. Lens Listing for Five-Lens Achromat
Surface
Radius
(mm)
Thickness
(mm)
Aspheric coefficients
Glass
 2 10 2 ,
 3 10 2 ,
 4 10 2 ,
 5 10 2 ,
mm-3
mm-5
mm-7
mm-9
-0.3036
-2.5530
-5.0378
-2.9428
6.9373
-7.0473
-7.3209
3.2878
OBJ:
Infinity
Infinity
Air
1
1.4784
0.6805
PMMA
2b
8.5921
0.1000
Air
STO:
Infinity
0.8018
Air
4
-0.9252
0.3805
PMMA
22.1007
-30.3701
63.5122
-9.6107
5
-1.7190
0.0500
Air
-6.3740
-9.7754
31.1070
-9.6795
6
1.5861
0.5560
PММА
-16.6813
2.2752
-2.1471
0.4165
7
2.0691
0.2697
Air
-0.2514
-5.4875
2.1004
-0.2941
8
2.1243
0.7675
PММА
-24.8607
11.2821
-2.3697
0.1701
9
2.1776
0.8068
Air
-18.1545
2.3803
-0.4134
0.0686
IMG:
Infinity
0.0000
a
3.7-mm FL
Diffraction Order and Diffractive lens phase coefficients: m =1; A1  -107 mm-2 ;
b
A2  103.9467 mm-4 ; A3  -1054.0420 mm-6 ; A4  4797.3402 mm-8 ;
A5  -11266.4930 mm-10 ; A6  14704.7480 mm-12 ; A7  -10400.9350 mm-14 ;
A8  3313.1630 mm-16 .
Ray and wave spherical aberration curves and a polychromatic modulation
transfer function curve for this microobjective are shown at Figure 3-6. The MTF
curve has been calculated in a paraxial image plane. As follows from the curves,
the resulting distorting plane-achromat with the relative aperture 1:2,4 provides the
resolution of 165 mm-1 across the field angle 2  60º with the contrast no lower
than 0,5, and 100 mm-1 with the contrast of 0,69. The residual chromatism in the
range from min  0,48613 μm to  max  0,65626 μm does not exceed 6,61 μm, and
the distortion is lower than  1%. Main rays fall onto the peripheral elements of
  0, 66  max , which is 19,8º.
the photodetector under the angle  max
Scheme assembling and apochromat optimization results
The studies have shown that apochromatization is achievable with an
additional DL included into the optical scheme of a microobjective; however the
scheme should satisfy several conditions.
First, alongside with positive lenses made of PMMA it should include at least
one negative lens made of PC that would be the place for a DL.
Second, the source optical scheme should provide a way to limit the focal
length drop within the selected spectrum range to the value equal to the spread of
the main intensity diffractive distribution maximum in the focal region along the
optic axis [12], even before the inclusion of a DL:
f   f ( max )  f ( min )  16 d K 2 ,
(1)
where K  is the (F-number). And the absolute value of Petzval surface’s radius,
that an apochromatic optical system could form a stigmatic image [12] on, should
satisfy the following condition (considering 2  =60○):
R  2,5 f  .
(2)
The above requirements alongside with the need to limit coma and astigmatism
could be fulfilled with a simple triplet, considering the acceptable surface
curvature from the point of ray intersection heights with the relative aperture and
field angle chosen above. In particular, this is stipulated by the fact that the starting
optical scheme doesn’t need the spherical aberration correction since it initially
supposes the inclusion of a DL and aspherizing of a refractor. However the
residual spherochromatism of the achromat based on a triplet notably limits the
resolution of a microobjective due to the extended spectrum range and enforces the
usage of a scheme with four RLs, each one with two aspherical surfaces. The
starting values of scheme’s construction parameters (refractor radiuses, lens
thickness and air gaps, and also parameters for even aspherical surfaces) can be
found using any known method of dimension or aberration calculation). However
the authors have used the pseudo-ray method that implies a ray tracing
approximation with different infinitesimal orders [13].
After placing a DL on the front surface of the second RL and the further
optimization, the final optical scheme and construction parameters of the 5-lens
apochromatic microobjective have been got (see Figure 7).
Figure 7. The optical scheme of the five-lens apochromat: 1– aperture stop; 2 – DL
A low level of monochromatic aberrations, a strict apochromatization with a
small tertiary spectrum, a virtually zero distortion and a wide spectrum range
polychromatic resolution not yielding to a 4-lens achromat resolution – these are
the main advantages of a 5-lens apochromat. Unfortunately, its main rays fall onto
  1,13 max  34º).
peripheral elements under unacceptably big angles (  max
It was possible to save the fulfillment of the above requirements and reduce the
limiting values ratio of field angles in the image and object domains at more than
  0,4 max ) and nearly made it telocentric ( max  12º) by
2,8 times (up to  max
changing three RLs of a microobjective with two-lens glued elements. At the same
time, the DL was moved onto the back refractor surface of the first glued element.
The optimization of this 8-lens scheme with only four aspherical surfaces has
allowed to build a plane-apochromat with the focal length f  =3,7 mm that
provides a resolution of 100 mm-1 with the relative aperture 1:2,4 and the contrast
no less than 0,5 within the field angle range 2  60º. The residual chromatism
does not exceed 6,9 μm in the range from  min  0,4 μm to  max  0,9 μm, the
transverse chromatism is comparable with the Airy disk and the distortion is less
than  1%. The optical scheme and the construction parameters of the eight-lens
plane-apochromat are shown at Figure 6 and Table 3 respectively.
Figure 6. The optical scheme of the eight-lens apochromat: 1– aperture stop; 2 –
DL.
Table 3. Lens Listing for Eight -Lens Apochromat
Aspheric coefficients
Surface
Radius
(mm)
Thickness
(mm)
Glass
OBJ:
Infinity
Infinity
Air
1
1.3380
0.5169
PC
2
1.0817
1.0762
Air
 2 10 2 ,
 3 10 2 ,
 4 10 2 ,
 5 10 2 ,
mm-3
mm-5
mm-7
mm-9
STO:
Infinity
0.2738
Air
4
-1.5219
0.9130
PC
5
-2.4540
0.3000
PMMA
6b
-1.8295
0.1500
Air
7
3.4708
1.6574
PMMA
8
-4.7064
0.3500
PC
9
-15.3523
0.1000
Air
10
14.6809
0.3500
PC
11
2.5013
2.6787
PMMA
12
-3.1078
2.1748
Air
IMG:
Infinity
0.0000
a
2.7333
7.1613
-10.6113
9.0267
3.9911
0.4239
0.0446
-0.0158
2.9526
-0.6118
0.0809
-0.0066
3.0736
0.3135
-0.0586
0.0045
3.7-mm FL
Diffraction Order and Diffractive lens phase coefficients: m =1; A1  -11.8913 mm-2 ;
b
A2  -13.1132 mm-4 ; A3  8.9955 mm-6 ; A4  -7.7903 mm-8 ;
A5  -0.6539 mm-10 ; A6  1.6548 mm-12 .
Ray and wave spherical aberration curves and a polychromatic modulation
transfer function curve for this microobjective, calculated in a paraxial image
plane, are shown at Figure 3-6. These curves clearly demonstrate the capabilities of
the eight-lens plane-apochromat.
Notice that the DL structures of all microobjectives presented in this work are
low-frequency and do not contain more than 10-15 ring zones. The low-frequency
of structures allows manufacturing them with a notched stroke profile or even twoply, which is possible with the same optical plastic – PMMA and PC. This
provides the nearly 100% effectiveness of a diffraction within the entire spectrum
range [10].
Conclusion
Summarizing the results of the present work, we can make the following
conclusions:
− An achromatization in the visible range with the low level of a secondary
spectrum is possible via the inclusion of a single DL to a microobjective having all
its RLs made of the same crown-like plastic;
− The inclusion of a single DL to a microobjective having its RLs made of two
marks of crown- or flint-like plastic allows an apochromatization in the wide range
of wavelengths, including the visual and the near IR ranges;
− Authors give recommendations that allow assembling the source schemes of
objectives and to calculate the construction parameters required for a following
optimization;
− The present work has shown that achievable optical characteristics of
achromatic and apochromatic microobjectives with plastic lenses satisfy the
qualifying standards for cell-phone objectives and CCTV-cameras.
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